Source optimization for image fidelity and throughput

ABSTRACT

A system and method for optimizing an illumination source to print a desired pattern of features dividing a light source into pixels and determining an optimum intensity for each pixel such that when the pixels are simultaneously illuminated, the error in a printed pattern of features is minimized. In one embodiment, pixel solutions are constrained from solutions that are bright, continuous, and smooth. In another embodiment, the light source optimization and resolution enhancement technique(s) are iteratively performed to minimize errors in a printed pattern of features.

CROSS-REFERENCE TO RELATED APPLICATION

The present application claims the benefit of U.S. ProvisionalApplication No. 60/541,335, filed Feb. 3, 2004, and which is hereinincorporated by reference.

FIELD OF THE INVENTION

The present invention relates generally to photolithographic processingtechniques and in particular to the optimization of an illuminationsource for printing a set of features on a wafer.

BACKGROUND OF THE INVENTION

In conventional semiconductor processing, circuit elements are createdon a wafer by exposing photosensitive materials on the wafer with apattern of transparent and opaque features on a mask or reticle. Theselectively exposed areas of the photosensitive materials can then befurther processed to create the circuit elements. As the size of thecircuit elements to be created on the wafer becomes similar to, orsmaller than, the wavelength of light or radiation that illuminates themask, optical distortions can occur that adversely affect theperformance of the circuit. To improve the resolution of thephotolithographic process, many circuit design programs utilize one ormore resolution enhancement techniques (RETs) that attempt to compensatefor the expected optical distortion such that the mask patterns will beprinted correctly on the wafer.

It is well known that one factor in determining how well a pattern offeatures on a mask will print is the pattern of light or radiation thatilluminates the mask. Certain types or orientations of features on amask will print with better fidelity when exposed with a particularillumination pattern. For example, off-axis illumination has been usedin microlithography for projection printing since the late 1980s becauseit increases resolution and depth of focus for certain layout patternsand design styles. Due to the demand to resolve smaller and smallerimages, the deployment of a variety of off-axis illumination sourceshapes was developed: first annular, then quadrapole, and lately dipole.These illumination source shapes can be formed by hard stop apertures orby diffractive optical elements (DOE). The latter is advantageousbecause it preserves light energy on the way from a laser source to themask (object) resulting in less throughput loss. In addition, DOEs canform very complex source shapes, with a smooth distribution of lightacross the aperture. This enables source tuning to print certain layoutfeatures with high resolution. Although lithographic exposure equipmentis compatible with the use of more complex illumination shapes, therehas been no technique to reliably determine a practical optimumillumination pattern for a given layout pattern, and in particular forthat layout pattern once RETs have been applied. Therefore, there is aneed for a method of determining what illumination pattern should beused for a particular pattern of features to be printed on a wafer.

SUMMARY OF THE INVENTION

To address the problems discussed above and others, the presentinvention is a method and apparatus for determining an optimumillumination pattern for use in exposing a mask or reticle having apattern of features thereon. In one embodiment, a design layout orportion thereof is analyzed and a mathematical relationship such as oneor more matrix equations are developed that relate how the features ofthe layout design will be printed from a light source having a number ofpixels with different intensities. The matrix equations are then solvedwith one or more matrix constraints to determine the intensity of thepixels in the light source that will produce the best possible imagingof the features on a wafer.

In another embodiment of the invention, the layout pattern used todetermine the optimum illumination pattern has had optical and processcorrection (OPC) or some other RET applied. The OPC corrected layout isused to determine the illumination pattern that can in turn be used torefine the OPC corrections in an iterative process.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a number of optimized illumination sources includinga pixilated source in accordance with the present invention;

FIGS. 2A and 2B illustrate a matrix equation used to determine anoptimized illumination source in accordance with one embodiment of thepresent invention;

FIG. 3 illustrates one embodiment of a computer system that may be usedfor implementing the present invention;

FIG. 4 illustrates how areas of a feature can be weighted whenoptimizing the illumination source and corrected during OPC;

FIG. 5 illustrates a representative pixilated illumination source;

FIG. 6 illustrates a series of steps to optimize a light source andperform a resolution enhancement technique on a design layout;

FIG. 7 illustrates a number of contact patterns A and B and 180 degreephase-shifted regions and corresponding diffraction diagrams;

FIGS. 8A and 8B are intensity profiles taken along horizontal cutlinesin the middle of the contact patterns A and B shown in FIG. 7;

FIG. 9 illustrates an optimized illumination pattern for an SRAM cellwith a uniform weighting and where the gates are weighted;

FIG. 10 illustrates optimized illumination patterns for a pattern offeatures with selective weighting, with OPC corrections and withselective weighting and corrections;

FIG. 11 illustrates smoothed versions of an original quadrapoleillumination D1 in order of decreasing smoothing;

FIG. 12 illustrates a graph of intensity along a cut line in a cell forsources C2 (pixel-based optimization) and C3 (parametric optimization);and

FIG. 13 illustrates various illumination patterns for an SRAM cellscaled to different sizes.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Before describing the illumination source optimizing techniques of thepresent invention, it is useful to provide an overview of previouslytried illumination optimization techniques. The previous lack ofrigorous formulations motivates discussion of the optimizationobjectives and constraints and the importance of using weighted andso-called Sobolev norms. As will be explained in detail below, thepresent invention states the main optimization problem as a set of theoptimization objectives in a form of functional norm integrals tomaximize image fidelity, system throughput, and source smoothness. Theseare reduced to a non-negative least square (NNLS) problem, which issolved by standard numerical methods. Examples of the present inventionare then provided for important practical cases including alternatingphase-shifting applied to regular and semi-regular pattern of contactholes, two types of SRAM cells with design rules from 100 nm to 160 nm,and complex semi-dense contact layer pattern. Finally, the presentinvention can be used with constraint optimization to smooth strongoff-axis quadrapole illuminations in order to achieve better imagefidelity for some selected layout patterns.

Methods for illuminator optimization can be classified by how the sourceis represented and how the objective function is defined. Table 1 belowlists these common applications for source optimization along with theirprincipal researchers, including parameterized, archels, and binarycontours based optimization, and gray-level pixel-based optimizationused in the present invention. The optimization objectives are listed inthe first column and include spectral fidelity, image fidelity, depth offocus, modulation, exposure latitude, and throughput. TABLE 1 OBJECTIVEFUNCTION AND SOURCE REPRESENTATIONS Objective/Source Parametized ArchelContour Pixel-based Spectral Fidelity Fehrs Mack Burkhardt ImageFidelity Vallishayee Present invention Depth of focus Ogawa RosenbluthBarouch** Present Smith Hsia invention** Brist Vallishayee InoueModulation Fehrs Burkhardt Inoue* Exposure Latitude Mack RosenbluthBarouch (Image slope or Smith NILS) Brist Throughput Hsia Presentinvention*DOF optimized by averaging through defocus**DOF optimized by off-focus optimization

The parameterized representation is used in source optimizations of theBrist and Bailey, Vallishayee, Orszag, and Barouch papers, and othersfrom the second column in the Table 1. FIG. 1 illustrates aparameterized approach 10 whereby source geometries are composed ofcircles, rectangles, and other primary shapes. Parameters of theseelementary shapes are subjected to an optimization procedure. Oneadvantage of this approach is a limited number of optimizationparameters. For example, for annular illumination only two parameters(sigma-in and sigma-out) are subjected to optimization. The drawback ofsource parameterization is that its optimization is considered as ageneric non-linear problem. This does not take advantage of the naturalstructure and properties of optical equations. In addition to this, thesolution domain is not full, e.g., is limited to those shapes that canbe parameterized and usually does not capture complex and/or gray-levelconfigurations. The parameterization can also be carried out by imposingangular or radial constraints on the source shape, or by consideringonly radial dependency as in the Inoue paper.

Diffraction pattern analyses and arch-based representations are used inthe Burkhardt paper. In the pupil diagram, important mask spectrumcomponents are isolated, then unit circles are drawn around them. Thesecircles break the source into arch-bounded areas, which are referred toas archels by analogy with the word pixels. The optimum source iscomposed of these archels 20 as shown in FIG. 1. In the Hsia paper,areas where only two circles intersect are used based on the 2-beamdesign principles for best focus latitude. In the Rosenbluth paper, thesource is broken into archels. Each archel is assumed to have a uniformbrightness. The brightness of the archels is found from the optimizationprocedure. An advantage of this method a natural division of the sourceinto regions that direct light into certain components of the pupilspectrum. The disadvantage is the assumption that within each archel thelight distribution is uniform. This is not necessarily true whennon-trivial pupil transmissions (with defocus for example) areconsidered.

A contour-based representation 30 as shown in FIG. 1 is described in theBarouch paper. Everything inside the contour is assumed to havebrightness 1 and outside a brightness of 0. This is a more compactrepresentation than the pixel-based one. The disadvantage comes from theusual problems of moving contours, e.g., contour self-intersections inthe form of swallow tails. Even a more important limitation is that onlybinary light sources are considered, like for the hard-stop apertures,and gray-level light distributions are not addressed.

To improve upon the prior attempts at source optimization, the presentinvention divides a source into a number of pixels and determines theoptimum brightness for each pixel for a given layout in a manner thatwill be physically practical to achieve and can be used in a real worldlithographic system. A pixel-based representation 40 as shown in FIG. 1can handle continuously-distributed light sources, like DOE produces. Itis the most flexible representation. However, the dimensionality of theproblem is the largest among all mentioned representations. Anoptimization solution for the fine-grained source may have more than10,000 pixels. However, with currently available computers a solutioncan be obtained.

Before discussing the particular mathematical techniques used tooptimize a light source in accordance with one embodiment of the presentinvention, it is useful to provide an overview of the techniquesemployed. FIGS. 2A and 2B illustrate the form of a linear equationsolved by an embodiment of the present invention in order to determinethe optimal distribution of light from an illumination source in orderto produce a desired image in accordance with the present invention. Asshown in FIG. 1, the area of a light source is divided into a number ofpixels 102 i, 102 ii, 102 iii, etc. The present invention thereforeserves to determine the proper intensity of the light source at eachpixel in order to optimally print a mask pattern on a wafer. As will bediscussed in further detail below, the illumination pattern is optimizedto print the desired mask pattern with optimal fidelity and to maximizethroughput of the system such that solutions with increased brightnessare favored over darker solutions. In addition, solutions that arecontinuous and smooth are favored over solutions that are discontinuousor solutions with bright spots that may damage the imaging optics of aphotolithographic printing system. In one embodiment, a continuoussolution means that no pixels having a non-zero intensity are surroundedby pixels having zero intensity, or stated another way, pixels havingnon-zero intensities are located in one or more groups of adjacentnon-zero pixels. In one embodiment, smoothness is defined such that theintensities of adjacent pixels do not vary by more than some predefinedamount.

The solutions for the distribution of light from the illumination sourceare generally symmetric around a central axis of the illuminationsource. However, a solution may not be symmetric for some featurepatterns.

As shown in FIG. 2A, the basic linear equation relates the product of atransfer matrix T and a source matrix R to a matrix I defining a desiredimage. The matrix T defines the contribution of each light source pixelto an image point on a wafer. The matrix R defines the intensity of thelight source at each pixel. The matrix I defines the desired pattern ofobjects to be created on a wafer. Typically, each entry in the matrix Iis either a 0 or a 1 defining areas of exposure or non-exposure on thewafer. Solving the matrix equation shown in FIG. 2A yields the entriesof the source matrix R that specify the desired intensity at each of thepixilated points in the light source. From the solution for the sourcematrix R, a diffractive optical element (DOE) can be fashioned for usewith a photolithographic printing system that will simultaneouslyproduce a pixilated light source whereby the pixels distribute the lightin the desired manner. Such diffractive optical elements can be made byDigital Optics Corporation of Charlotte, N.C., the details of which areconsidered known to those of ordinary skill in the art. Alternatively, adedicated light source can be produced that will distribute illuminationlight in the designed pattern using conventional optical techniques,such as lenses and stops, etc.

FIG. 3 illustrates a representative computer-based system forimplementing the present invention. The computer-based system includes acentralized or distributed computer 150 (i.e., a computer with multiplemicroprocessors, or a network of linked computers) that reads at least aportion of a layout design to be formed on an integrated circuit orother device. Typically, the layout design is stored in a layoutdatabase 152. The computer 150 reads a computer program stored on acomputer-readable medium 154, such as a CD-ROM, DVD, magnetic tape, harddisc drive, flash card, etc., or may be received via a wired or wirelesscommunication link. The computer system 150 executes the instructions ofthe computer program in order to optimize the illumination pattern froma light source 160 that will be used with a photolithographic mask orreticle 170 to produce the desired features on a semiconductor wafer 192using a suitable imaging system. Typically, such an imaging system willbe a 4× reduction system, i.e., the dimensions on the mask will be 4times larger than the corresponding image features on the wafer.However, other reduction factors, such as 1×, 5×, 6×, or even 10×imaging systems can be used with this invention as well. In oneembodiment of the invention, the pattern of illumination is controlledby a diffractive optical element 190 positioned between the light source160 and the mask 170. However, the present invention could also be usedto design the light source 160 itself such as by adding hard stopelements to the light source or other optical elements required toproduce the distribution of light determined by the computer system 150.

The computer system 150 may also perform one or more resolutionenhancement techniques on the layout design such as optimal proximitycorrection (OPC) in order to produce an OPC corrected mask data that isprovided to a mask writer 200. The OPC corrected mask data may beprovided to the mask writer 200 on a computer-readable um 210 such as aCD-ROM, DVD, hard disc, flash card, or the like. Alternatively, the OPCcorrected mask data may be provided to the mask writer 200 via awireless or wired communication link 220. In one embodiment of theinvention, the computer system 150 that determines the distribution ofillumination light resides within the United States. However, it ispossible that the computer system 150 may communicate with one or moreremotely located computers 250 that may be outside the United States.Data is transmitted to the one or more remote computer systems 250 via awired or wireless communication link, such as the Internet 260. Theremote computer system 250 performs the illumination source optimizationmethod of the present invention and the results of the optimizationmethod are used to produce the light source 160 or a diffractive opticalelement 190 used to print the mask pattern 170 on one or more wafers180.

As will be discussed in further detail below, the present inventionoptimizes an illumination source by reading all or a portion of adesired layout pattern such as that shown in FIG. 4. Here, a layoutpattern 280 includes a number of polygons 282, 284, 286, 288, 290, thatdefine objects to be created on a semiconductor wafer or may defineobjects to assist in the printing of objects such as subresolutionfeatures, phase shifting regions, etc. In conventional processing, eachof the polygons 282-290 is fragmented with a number of fragmentation endpoints 294 that divide the perimeter of each polygon into a number ofedge segments. As will be appreciated by those of ordinary skill in theart, resolution enhancement technologies, such as OPC, operate toincrease the ability of the layout pattern to print correctly by, forexample, moving each edge segment inwards or outwards, addingsub-resolution features or phase shifters, serifs, etc., such that thepattern of polygons will be correctly printed on a wafer. As will beexplained in further detail below, in one embodiment of the invention,the distribution of light from the light source is optimized such thateach polygon in the desired layout that is supposed to print on thewafer will print equally as well. However, in other embodiments of theinvention, it may be desirable to emphasize certain portions of thelayout that are critical for a circuit operation. For example, FIG. 4shows two areas 300 that may correspond to transistor gate regionswhereby accurate formation of the gate areas on a semiconductor wafer iscritical to circuit operation. In some embodiments, the areas 300 areweighted in the matrix calculation described above such that thedistribution of light from the light source is optimized to accuratelyprint the weighted regions with greater fidelity at the expense of thefidelity in the other regions.

FIG. 5 shows a representative light source that is optimized to print aset of features on a semiconductor wafer. The light source 40 is dividedup into a number of pixels 102 i, 102 ii, and 102 iii, etc. Each pixelin the light source is assigned a brightness in accordance with thetechniques described below. In the example shown in FIG. 5, the lightsource is generally symmetric about a center axis 310 of the lightsource. However, this is not required. The distribution of light in thelight source is generally produced using a diffractive optical elementthat operates as a hologram to produce the desired light pattern from anincoming light source such as a laser or other coherent radiationsource. The use of diffractive optical elements is currently preferredbecause the light source can be readily changed for other distributionpatterns in order to expose other patterns of features on a wafer.

FIG. 6 illustrates one possible sequence of steps in which the presentinvention can be used. Beginning at a step 400, a layout database or aportion thereof is retrieved by the computer system. The pattern offeatures from the database may form a target layer that defines thedesired pattern to be created on a wafer. In other embodiments, thepattern of features may be modified by other tools, such as a designrule checker (DRC), etc., to create the target layer. At a step 402,initial resolution enhancement techniques (RETs) such as OPCcalculations are performed assuming an initial distribution of lightfrom the light source. At a step 404, the distribution of light from thelight source is optimized according to the methods set forth in furtherdetail below. The mask layout pattern used to optimize the light sourcewill be the OPC corrected layout. At step 406, it is determined if thesimulated results of the printed layout using the source optimized instep 404 are within set tolerances compared to the target layer. If not,processing proceeds to step 408, whereby additional OPC calculations areperformed using the new optimized light source. This creates a revisedOPC corrected layout. Processing can then return to step 404, wherebythe light source illumination pattern is further optimized to print therevised OPC corrected layout data. Alternatively, processing couldproceed from step 408 to step 406 and additional OPC calculationsperformed in a loop until it is determined that the layout will printwith the desired tolerances. Once the answer to step 406 is YES, theprocessing can be finished.

By using the revised OPC corrected layout data as the mask layout, themask layout and light source illumination pattern can be iterativelyrefined to ensure accurate printing of the desired feature pattern on awafer. Although the flow diagram shown in FIG. 6 shows performing a RETbefore optimizing the light source distribution, it will be appreciatedthat light source optimization could be performed using an uncorrectedlayout description first and then using the optimized light sourcedistribution to perform the RETs.

The source intensity is a 2D, non-negative, real valued function, whichis defined inside a circle of radius σ (partial coherency). If aCartesian coordinate system is positioned in the center of this circle,then the source can be represented as the functionS=S(x,y)≧0,x ² +y ²≦σ².  (1)Discretization of this yields a pixel-based source representation.S _(s) =S(x _(i) ,y _(j))≧0,x _(i) ² +y _(j) ²≦σ²,  (2)where the size Δx=Δy of the pixels is dictated by the number n of thediscretization intervals between −σ and σ: $\begin{matrix}{{\Delta\quad x} = {{x_{i + 1} - x_{i}} = {{\Delta\quad y} = {{y_{i + 1} - y_{i}} = {\frac{2\quad\sigma}{n}.}}}}} & (3)\end{matrix}$the total amount of energy (per time unit) falling onto the mask is$\begin{matrix}{{E = {{\underset{A}{\int\int}{S\left( {x,y} \right)}\quad{\mathbb{d}x}{\mathbb{d}y}} \equiv {S}_{1}}},} & (4)\end{matrix}$where the operator ∥·∥₁, is a Manhattan functional norm l₁, and A is asource area. The throughput of the printing system is dictated by thisenergy. Among otherwise equally fit sources, generally preferable is asource with the largest throughputE(S)=∥S∥ ₁→max.  (5)

Another requirement for the source design is to avoid sharp spikes thatcan damage lenses of the photolithographic system, and generally to keeplight evenly distributed across the source. This requirement can beexpressed through a constraint $\begin{matrix}{{{\max\limits_{x,y}\quad{S\left( {x,y} \right)}} \leq S_{\max}},} & (6) \\{or} & \quad \\{{{S}_{\infty} \leq S_{\max}},} & (7)\end{matrix}$which limits the source energies to some value S_(max) that can betolerated by lenses; the operator ∥·∥_(∞) is the Chebyshev or infinityfunctional norm l_(∞). A combination of Equation 5 and Equation 7constitutes a constrained optimization problem∥S∥₁→max∥S∥_(∞)≦S_(max),  (8)which has an obvious solution S(x, y)=S_(max), meaning that the sourceis uniformly lit. Equation 8 is more relevant to optimization of thesource formed by hard stop apertures than by DOEs. DOE redirects lightrather than blocks it, so the energy in Equation 5 does not depend onthe shape of the source, but on the power of the laser. In this case therelevant formulation is∥S∥₁=E₀∥S∥_(∞)→min,  (9)which means that source distributions are restricted to those that areformed by the same power supply. Thus the energy E falling onto the maskis fixed (E=E₀), and the impact of possible spikes or non-uniformitiesshould be minimized. The optimization of Equation 9 has the solutionS(x, y)=E₀/A, where A is the total area of the source. Indeed, ∥S∥_(∞)is limited by E₀/A becauseE ₀ =∥S∥ ₁ ≡∫∫Sdxdy≦S _(max) A≡∥S∥ _(∞) A.  (10)

In other words, it is not possible to do better in ∥S∥_(∞) minimizationthan to reach ∥S∥_(∞)=E₀/A. This limit is reached when S=E₀/A, so S(x,y)=E₀/A solves Equation 9. This is the same solution as for Equation 8,if we match constants S_(max)(x, y)=E₀/A.

Though Equations 8 and 9 have the same solutions (uniformly litillumination pupil), it does not necessarily mean that they will havethe same effect when added to a larger optimization problem of thepattern transfer fidelity.

The l_(∞) norm used in Equation 10 can be replaced by the Euclidean l₂norm∥S∥ ₂ =[∫∫S ²(x,y)dxdy] ^(1/2),  (11)which is an optimization problem. It is also less harsh in penalizingintensity spikes, which is a desirable property considering that somenarrow spikes can be tolerated or mitigated by lowering the surroundingenergies. Similar to Equation 10, it can be shown that the resultingoptimization problem∥S∥₁=E₀∥S∥₁→min  (12)is solved by a uniform distribution S(x,y)=E₀/A.

Non-uniformity of the source intensity accelerates degradation ofreflective and refractive elements in the optical path as far as thecondenser lens. The lens coating is especially sensitive to the laserirradiation and can suffer loss of transmission. It is not uncommon todiscover during hardware maintenance that the source shape has becomeburned into the lens coating. However, it is hard to quantify potentialdamage from different source shapes other than to say that the light hasto be evenly and smoothly—in some sense—spread across the illuminationaperture.

In addition to variations in the formalization of the requirement thatare represented in Equations 9 and 12, a useful generalization comesfrom utilization of so-called Sobolev norms. These norms compare notonly values of the functions but also values of their derivatives.Considering only the Euclidean type of Sobolev norms and restricting thecomparison to the first and second derivatives, the Sobolev metric∥·∥_(sob) is calculated as follows:∥S∥ _(sob)=[α₀ ² ∥S∥ ₂ ²+α₁ ² ∥L ₁ S∥ ₂ ²+α₂ ² ∥L ₂ S∥ ₂ ²]^(1/2),  (13)where α₀, α₁, α₂, are metric constants, L₁ is an operator of the firstderivative, and L₂ is an operator of the second derivative. Varying themetric constants, source smoothing is achieved by lowering the intensityvariability, and/or lowering the first derivatives, and/or lowering thesecond derivatives. Though not all combinations of metric constants makesense: if α₀=α₁=0, α₂>0, then the following minimization problem arisesin the Sobolev metric∥S∥₁=E₀∥S∥_(sob)→min.  (14)

It yields, for example, a non-uniform linear solution S(x, y)∝2+x+y.This intensity distribution is smooth, but does not evenly spread lightacross the source. Thus, it is reasonable to limit the metric constantsto those that satisfyα₀ ²+α₁ ²>0,  (15)

Under the conditions of Equation 15 the minimization problem of Equation14 has the same solution: S(x,y)=E₀/A as in Equations 12 and 9. Theproblem of Equation 12 is a special case of Equation 14 when α₀=1,α₁=α₂=0. Equation 14 is a part of the general optimization problem inaddition to the image fidelity objective.

The pixel-based source representation can naturally be used insatisfying Equation 14. Notions of evenly or smoothly lit source do notfit into the frameworks of contour-based representations 30 orarch-based representations 20 shown in FIG. 1.

For dense gratings normalized image log slope (NILS) is proportional tothe number of captured diffraction orders. This indicates that spectralfidelity as an optimization metric relates to NILS and thus to theoptimization of exposure latitude.

Image quality can be judged by modulation (or Michelson contrast)$\begin{matrix}{M_{c} = {\frac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}}.}} & (16)\end{matrix}$

The maximum modulation may be achieved by choosing to light thoseregions on the source that shift the important components of the maskspectrum into the pupil. Similarly, simulated annealing may be used tooptimize radially-dependent sources. The shortcoming of this objectiveis that the modulation as a metric of image quality is relevant only tosimple gratings or other highly periodic structures. For phase shiftingmasks (PSM), one can achieve maximum modulation of 1 just by capturingtwo interfering +/−1 orders in the pupil, which zeros I_(min). However,this does not faithfully reproduce mask features because high spectralcomponents are ignored. Equation 16 is relevant for simple harmonicsignals, where it serves as a measure of signal-to-noise ratio. It isquestionable for judging printability of complex patterns, or evenisolated lines, with Weber contrast W_(c)=(I_(max)−I_(min))/I_(min)being a better metric.

The image fidelity is a more universal metric than modulation. Toestablish this metric, we can start with the notion of the layout data(or OPC corrected layout data) layer, which represents the desiredpattern on the wafer. For this layer we can build a characteristic 2Dfunction, which is 1 inside the layer shapes and 0 outside. Thisfunction is an ideal image, or an ideal distribution of the lightintensity on the wafer,I _(ideal) =I _(ideal)(x,y).  (17)

The ideal image can also be expressed through the complex-valued masktransmission function m(x,y) asI _(ideal) =m(x,y)m*(x,y),  (18)where the asterisk denotes a complex conjugation.

The optimization objective F can be formed as a Euclidean norm l₂ of thedifference between the real I(x,y) and ideal images:F=∥I−I _(ideal)∥₂ ={∫∫[I(x,y)−I _(ideal)(x,y)]² dxdy} ^(1/2).  (19)

F is called image fidelity. As an optimization objective, this integralwas first described by Vallishayee and called contrast. Using theParceval theorem, which states that l₂ norms are equal in the space andin the frequency domains, Equation 19 to the frequency domain:$\begin{matrix}\begin{matrix}{F = {{I - I_{ideal}}}_{2}} \\{= {{{\hat{I}\left( {k_{x},k_{y}} \right)} - {{\hat{I}}_{ideal}\left( {k_{x},k_{y}} \right)}}}_{2}} \\{{= \left\lbrack {\sum\limits_{i,j}{{{\hat{I}\left( {k_{xi},k_{yj}} \right)} - {{\hat{I}}_{ideal}\left( {k_{xi},k_{yj}} \right)}}}^{2}} \right\rbrack^{1/2}},}\end{matrix} & (20)\end{matrix}$where k_(x), k_(y), are spectral coordinates; i, j are summation indicesof the discrete spectrum; the circumflex denotes a Fourier transform.The equality in Equation 20 means that the image and spectral fidelitiesare the same metrics when expressed in Euclidean norm.

An off-axis illuminator design is often conducted in the spatialfrequency domain. In the frequency domain, image intensity for apartially coherent system and a periodic mask transmission is defined bythe Hopkins summation $\begin{matrix}{{{\hat{I}\left( {k_{xi},k_{yj}} \right)} = {\sum\limits_{f,g}{{\hat{h}\left( {f,g,{k_{xi} + f},{k_{yj} + g}} \right)}\quad{\hat{m}\left( {f,g} \right)}\quad{{\hat{m}}^{*}\left( {{k_{xi} + f},{k_{yj} + g}} \right)}}}},} & (21)\end{matrix}$where ĥ=ĥ(f, g, p, q) are transmission cross-coefficients (TCCs). In thefrequency domain, the ideal image can be obtained from Equation 18 usingthe Borel convolution theorem to convert the multiplication to thefollowing convolution: $\begin{matrix}{{\hat{I}}_{ideal} = {\sum\limits_{f,g}{{\hat{m}\left( {f,g} \right)}\quad{{{\hat{m}}^{*}\left( {{k_{xi} + f},{k_{yj} + g}} \right)}.}}}} & (22)\end{matrix}$

Subtracting Equations 21 and 22, the expression for the spectralfidelity is in the form:F=∥Î−Î _(ideal)∥=∥Σ(ĥ−1){circumflex over (m)}{circumflex over(m)}*∥.  (23)

This expression can be minimized by attempting to setting components ofĥ to 1. For the high-frequency components of the mask transmission thisis an unattainable goal, because the optical system is band-limited andall the correspondent high-frequency TCCs must necessarily be 0. Thus,only a limited number of TCCs can be controlled, which means that thehigh-frequency elements can be removed from the sum (4) and an objectivefunction in the form of a truncated summation considered. In thecanonical optical coordinates and for a clear circular unaberratedpupil, each TCC value ĥ(f, g, p, q) is the area of intersection of twoshifted pupils (unity circles) with centers at (f, g) and (p, q), and asource area A, normalized by the source area. Thus, ĥ(f, g, p, q) isequal 1 when the source area is fully encircled by both pupils. Usingthis simple geometrical consideration, a few elements (a few orders) canbe “hand-picked” from the truncated sum of Equation 21 to find thesource area as a combination of intersections of correspondent unitycircles, or combination of archers. In a more rigorous way, the sum ofEquation 21 can be rewritten in a matrix form and minimized to find ĥ(f,g, p, q), then the source can be constructed out of archels.

In the spatial domain, it is often beneficial to consider the followingweighted image fidelity errorF _(w) =∥I−I _(ideal)∥_(w2) =∥{square root}{square root over (w)}·(I−I_(ideal))∥₂ =[∫∫w·(I−I _(ideal))² dxdy] ^(1/2).  (24)where the weighting function w=w(x, y) is formed to emphasize importantdesign features and regions (gates, landing pads, etc.). The weightingfunction can be formed in such way as to effectively make imagecomparison one-dimensional by using a 2D characteristic function, whichequals 0 everywhere except some ID “cutlines” where it is infinite. Inthis case image fidelity in Equation 24 becomes a ID integral in theformF _(1D) =[∫∫[I(z)−I _(ideal)(z)]² dz] ^(1/2),  (25)where coordinate z is a distance along the cutline. Comparison of imagesalong a cutline or multiple cutlines simplifies the optimization problemand speeds up computer calculations at the expense of comprehensivenessand perhaps accuracy of some 2D feature reproductions. Cutlines havebeen used to maximize the focus latitude given fixed exposure latitude.

Image fidelity can be expressed in other than l₂ norms. If the Chebyshevnorm l_(∞) $\begin{matrix}{F_{\infty} = {{{I - I_{ideal}}} = {\max\limits_{x,y}{{I - I_{ideal}}}}}} & (26)\end{matrix}$is used for this purpose, then the optimization minimizes the maximumdifference between ideal and real images, rather than the averagedifference, as in Equation 24. Equation 26 is a justifiable metric,because printing limits are dictated by the regions of the worstprintability, e.g., those areas where ideal image reproduction is theworst and the maximum difference between ideal and real images isobserved. Chebyshev fidelity in Equation 26 has two drawbacks. First, itis harder to minimize l_(∞) than l₂. Difficulty grows with the number ofgrid points used in computer simulations for I. The solution may not beunique or slow convergence is observed. Second, the Chebyshev norm isnot equivalent to any metric in the frequency domain, as in Equation 24.Relationships between spectral and frequency norms are governed by theHausdorff-Young inequality. It states that the following inequalityholds between norm l_(p)(1≦p≦2) in the space domain and norml_(q)(q=p/(1−p)) in the frequency domain∥Î∥_(q)≦∥I∥_(p).  (27)

When p=1, q=∞, it is inferred that the Manhattan norm fidelity l₁, inthe space domain limits the Chebyshev fidelity in the frequency domain∥Î∥_(∞)≦∥I∥₁.

However, the inverse is not true, so minimization of Equation 26 doesnot limit any fidelity error in the frequency domain.

A useful generalization of the fidelity metrics of Equations 24 and 26can be achieved by considering Sobolev norms. For practical purposes thecomparison can be limited to the first derivative only. Using linearcombination the first derivative and Equation 24, Sobolev fidelity(squared) isF _(sob) ²=α₀ ² ∥I−I _(ideal)∥²+α₁ ² ∥L ₁ [I−I _(ideal)]∥².  (28)The metric coefficient α₁, defines the weight for the image slopefidelity. Equation 28 states that ideal and real images are close toeach other when their values are close and the values of their firstderivatives are close.

An important practical case of Equation 28 is when α₀=1, α₁=1, e.g.,when only first derivatives are compared in the F_(sob) metric. Thefirst derivative of the ideal image is 0 almost everywhere in the waferplane except a thin band around edges of the target layer where thisfunction is infinite (or very large, if the ideal image is slightlysmoothed). Under these conditions minimization of F_(sob) is equivalentto the maximization of the slope of the real image in a thin band, whichrelates to the problem of maximization of the exposure latitude. Withoutbeing formally stated, this objective—expressed in the norm l_(∞)—isused herein.

To optimize for the best process window, rather than for exposurelatitude only, it is important to account for defocus effects in theobjective function, which is usually done by averaging it through focusvalues f_(k), so that instead of Equation 24 the following is optimized$\begin{matrix}{{F_{w}^{f} = {\sum\limits_{k}F_{w}^{f_{k}}}},} & (29)\end{matrix}$where F_(w) ^(f) ^(k) is the image fidelity calculated for focus f_(k).The averaging can be carried out for only two values, for some “plus”and “minus” defocus positions so thatF _(w) ^(f) =F _(w) ⁺ +F _(w) ⁻.

To reduce optimization run time further, an approximate condition F_(w)⁺≈F_(w) ⁻, and carry optimizations off-focusF _(w) ^(f) =F _(w) ⁺ =∥I ⁺ −I _(ideal)∥_(w2),  (30)Here I⁺ is an off-focused wafer image. Numerical experiments show thatEquation 30 out of focus and Equation 29 averaging optimization resultsare hard to differentiate, so that the run time overhead of theaveraging technique is not justified. However, the results exhibit astrong dependence on the plus defocus position, so that in eachapplication careful exploration of this value should be conducted,guided by considerations for the required or expected depth of thefocus. In the examples described below, half of the focal budget for the“plus” defocus position is used.

The optimization objective of source smoothness of Equation 14 arecombined with the weighted off focus image fidelity of Equation 30 arecombined to state the following optimization problem:S≧0∥S∥₁=E₀∥S∥_(sob)→minF _(w) ⁺ =∥I ⁺ −I _(ideal)∥_(w2)→min  (31)

It is convenient to introduce the normalized source intensity$r = {\frac{S}{E_{0}}.}$Then conditions in Equation 31 can be expressed using this normalizedquantity asr≧0∥r∥₁=1∥r∥_(sob)→min∥I ⁺ −I _(ideal)∥_(w2)→min

This problem has two mutually exclusive minimization objectives that arecombined in some proportion γ to state a correct minimization problem.This leads to the following formulationr≧0∥r∥₁=1γ² ·∥I ⁺ −I _(ideal)∥_(w2) ²+(1−γ)² ·∥r∥ _(sob) ²→min  (32)

The optimization proportion 0≦γ≦1 balances two objectives, the imagefidelity and source smoothness. When γ=1 the image fidelity is optimizedalone, and at the other extreme at γ=0 a trivial problem of smoothingthe source without paying attention to the image fidelity is obtained.Equations 32 constitute a constrained quadratic optimization problem forthe normalized source intensity r=r(x, y).

The solution of Equation 32 is reduced to a sequence of non-negativeleast square (NNLS) optimizations using the Courant style reduction of aconstrained to an unconstrained problem, and subsequent discretizationof the source and image intensities. Multiplying the equality conditionin Equation 32 by a large positive number, C_(n), and adding it to theminimization objective, results inr≧0γ² ·∥I ⁺ −I _(ideal)∥_(w2) ²+(1−γ)² ·∥r∥ _(sob) ² +C _(n) ²·(∥r∥₁−1)²→min  (33)

This is solved for a sequence of increasing C_(n) values, which forcesthe error of the constraint ∥r∥₁=1 to be sufficiently small for somelarge n.

The minimization objective in Equation 33 is a functional of the sourcer=r(x, y). To find a form of this dependency we use the Abbe approach todescribe the imaging system. Consider a spherical wave of amplitudea_(s) coming from the source point {right arrow over (k)}_(s)=(k_(x),k_(y)). The source intensity at this point is S_(s)=a_(s)a^(*) _(s).This wave is incident to the object as a plane wavea _(oi) =a _(s) exp(i{right arrow over (k)}{right arrow over (x)}).

This amplitude is modulated by the mask, so that the transmittedamplitude isa _(ot) =a _(s) exp(i{right arrow over (k)} _(s) {right arrow over(x)})m,where m is a complex transmission of the mask. Complex amplitude a_(p)that is arriving at the pupil plane is the Fourier transform of theamplitude a_(ot) in the object plane:a _(pi) =F[a _(ot) ]=a _(s) F[m exp(i{right arrow over (k)} _(s) {rightarrow over (x)})].

This is multiplied by the pupil function, so that the transmittedamplitude is:a _(pt) =a _(pi) P=a _(s) F[m exp(i{right arrow over (k)} _(s) {rightarrow over (x)})]P.  (34)

The image of the object is then formed at the image plane by inverseFourier transformation, so thata _(i) =F ⁻¹ [a _(pt) ]=a _(s) F ⁻¹ [F[m exp(i{right arrow over (k)}_(s) {right arrow over (x)})]P].  (35)

By applying the shift theorem for the Fourier transformation, the resultisF[m exp(i{right arrow over (k)} _(s) {right arrow over(x)})]={circumflex over (m)}({right arrow over (k)}−{right arrow over(k)} _(s)),where {circumflex over (m)}=F[m] is the Fourier transform of the mask.With this, the amplitude at the image plane isa _(i) =A _(s) F ⁻¹ [{circumflex over (m)}({right arrow over (k)}−{rightarrow over (k)} _(s))·P].  (36)

The shift theorem is again applied to the inverse Fourier, resulting ina _(i) =a _(s) exp(i{right arrow over (k)} _(s) {right arrow over (x)})F⁻¹ [{circumflex over (m)}·P({right arrow over (k)}+{right arrow over(k)} _(s))].  (37)

The light intensity in the image plane is a sum of the amplitude modulesnormalized to the source energy $\begin{matrix}{I = {\sum\limits_{s}{a_{i}{a_{i}^{*}/{\sum\limits_{s}{a_{s}a_{s}^{*}}}}}}} & (38) \\{\quad{= {\sum\limits_{s}{a_{s}^{2}{{{F^{- 1}\left\lbrack {\hat{m} \cdot {P\left( {\overset{\rightarrow}{k} + {\overset{\rightarrow}{k}}_{s}} \right)}} \right\rbrack}}^{2}/{\sum\limits_{s}{a_{s}{a_{s}^{*}.}}}}}}}} & \quad\end{matrix}$

The Abbe formula of equation 38 is rewritten in a convolution form.Starting with equation 37 and apply the Borel convolution theorem,yieldsF[fg]=F[f]{circle over (×)}F[g].  (39)

This yieldsa _(i) =a _(s) exp(i{right arrow over (k)} _(s) {right arrow over (x)})F⁻¹ [{circumflex over (m)}·P(k+k _(s))]=a _(s) exp(i{right arrow over(k)} _(s) {right arrow over (x)})m{circle over (×)}(F ⁻¹[P]·exp(−i{right arrow over (k)} _(s) {right arrow over (x)})).  (40)

Introducing Abbe kernelsK _(s) =F ⁻¹ [P]·exp(−i{right arrow over (k)} _(s) {right arrow over(x)}),  (41)the image plane amplitude can be represented as a convolution operationa _(i) =a _(s) exp(i{right arrow over (k)} _(s) {right arrow over(x)})m{circle over (×)}K _(s).  (42)

Using pointwise summation over the source produces the followingexpression for the image intensity $\begin{matrix}{I = {\sum\limits_{s}{a_{s}a_{s}^{*}{{{m \otimes K_{s}}}^{2}/{\sum\limits_{s}{a_{s}a_{s}^{*}}}}}}} & (43) \\{\quad{= {\sum\limits_{s}{S_{s} \cdot {{{m \otimes K_{s}}}^{2}/{{S}_{1}.}}}}}} & \quad\end{matrix}$

The convolution form of the image integral speeds up calculations whenbeing used with the lookup table approach. Using constraint ∥S∥₁=E₀ fromequation 12, the image intensity in equation 43 can be expressed throughnormalized source intensities $\begin{matrix}{I = {\sum\limits_{s}{r_{s} \cdot {{{m \otimes K_{s}}}^{2}.}}}} & (44)\end{matrix}$

The linearity of image intensity as a function of source pixelssimplifies the solution of equation 33, the minimization can be reducedto solving—in a least square sense—a system of linear equations. Todeduce this system, the image on a wafer is discretized and all imagepixels are sequentially numbered, which yields the image vector I andallows us to express equation 44 in a matrix formI=Tr,  (45)where the source vector r={r_(s)} and the components of thetransformation matrix T can be computed from the convolutions inequation 44. Equation 45 can be substituted into equation 33, resultingin the following optimization problem for the source vector rr≧0∥Gr−a∥ ₂→min  (46)where matrix G and vector a consist of the following blocks$\begin{matrix}{{G = \begin{pmatrix}{\gamma\sqrt{w}T} \\{\left( {1 - \gamma} \right)\alpha_{0}E} \\{\left( {1 - \gamma} \right)\alpha_{1}L_{1}} \\{\left( {1 - \gamma} \right)\alpha_{2}L_{2}} \\C_{n}\end{pmatrix}},\quad{a = \begin{pmatrix}{\gamma\sqrt{w}I_{ideal}} \\0 \\0 \\0 \\C_{n}\end{pmatrix}}} & (47)\end{matrix}$

The optimization problem of equation 46 is a standard NNLS problem, withwell-established methods and software packages to solve it, including aMATLAB routine NNLS. Equation 46 is solved for a sequence of increasingvalues of C_(n) until the condition ∥r∥₁=1 is satisfied with requiredaccuracy.

EXAMPLE 1 Periodic Array of Contacts, Alternating PSM

Geometries and process conditions for our first two examples areborrowed from the Burkhardt paper. Contact patterns A and B are shown inFIG. 7. They are composed of clear and 180° phase shifted, 210 nmcontact holes that are imaged on a dark background using λ=248 nm andNA=0.5. Contact pattern A has 640 nm pitch in the γ and 320 nm pitch inthe x direction. Contact pattern B is less regular, with a basic pitchof 320 nm in both directions. Pupil diagrams are shown in the secondcolumn of FIG. 7. Positions of diffraction orders are marked by smallsquares, with their brightness being proportional to the orderamplitudes. Thick white circles show the source area σ=0.6.

For the pattern A, four light squares represent the major diffractionorders that are larger than the four minor diffraction orders, marked asthe darker squares and positioned farther from the x-axis. Around eachof the eight shown diffraction orders a unit circle is erected to show ashifted pupil. These circles break the source of size σ=0.6 into 27archels, including 1, 2, and 3. Archels 1 and 2 maximize TCCs betweenthe four major orders, while archel 3 maximizes TCCs between four minorand four major orders. Inset A1 shows the optimum source when thebalance parameter γ is large, γ=0.91. The source mainly consists of abright central archel 1 in a rhombic shape. When γ is small and sourcesmoothing is increased (γ=0.07, see inset A2 of the FIG. 7), lightspreads from the central archel and an increase in brightness of archels2 and 3, resulting in a source shape somewhat reminiscent of thehandwritten letter x.

Four major diffraction orders and their 9 archels for the pattern B areshown in a pupil diagram in the second row of the FIG. 7. Optimum sourceB1 for γ=0.91 is a vertical dipole, mainly formed by archels 4 and 5.When source throughput and smoothness are increased at γ=0.07, twoadditional archels 6 and 7 are lit.

We showed four optimal source designs A1, A2, B1, and B2. In theoriginal Burkhardt study, only two of them, similar to A1 and B2, werefound and analyzed. This suggests that a comprehensive optimization ofthis study or a similar one is required even for highly periodicpatterns, or else some advantageous designs may be overlooked.

EXAMPLE 2 SRAM Design 1, Binary Mask

In this section we consider a 130 nm SRAM design from the Brist paper.Geometry of this SRAM cell is much more complicated than for thecontacts from the previous section and cannot be tackled by a simpleanalysis of diffraction orders. The cell from FIG. 9 tiles large regionof the SRAM design in a way that is called in geometry wallpapers tilinggroup pmm. A fundamental cell of this tiling consists of four mirroredpatterns of FIG. 9, with symmetries along the vertical and horizontalaxis. These symmetries reduce the solution of the optimization problemto one quadrant of the source, with subsequential mirroring of resultsalong vertical and horizontal axes. Simulation runtime can be reducedfarther by simulating one-forth of the fundamental cell, then filteringits spectrum to induce even periodic boundary conditions in both thevertical and horizontal directions. The optimization domain is chosen tobe a circle of σ=0.8 and the balancing parameter γ is 0.38.

Two different weighting styles for the image fidelity (24), including auniform edge weighting and a gate weighting, are compared. For theuniform edge weighting, a layer that covers the edges of the polygons asa 40 nm wide uniform band is created. Image fidelity is weighted 36times larger inside this band than outside. The correspondent optimizedsource, C1, is shown in the first row of FIG. 9. C1 consists of 12peripheral poles lying inside the annulus 0.7<σ<0.8. The gate weightingis shown in the second row of the same figure. The gate weighting layer(weight 36) covers six gate regions, where it is important to controlfidelity more accurately than in the rest of the pattern (weight 1).Changes in the weighting style substantially affected sourceconfigurations: the two vertical poles disappear, while the remainingten poles spread evenly across the periphery. In addition to the main 10poles, a weak quadrapole can be seen, with two horizontal poles atσ≈0.65 and two vertical poles at σ≈0.45.

EXAMPLE 3 SRAM Design 2, Attenuated Mask

In this example the SRAM pattern and process conditions are similar tothe Barouch paper. We optimize the source for the SRAM cell shown inFIG. 13 with the purpose of comparing contour-based, as in the Barouchpaper, and pixel-based optimization results. A uniform weighting styleis chosen with the weight 32, balance parameter γ is 0.38, and opticalconditions are λ=248 nm, NA=0.5, σ=0.8.

Optimizations were run for the scaled designs with 140 nm, 160 nm, 180nm, 200 nm, 220 nm, and 250 nm feature sizes. The resulting sourceconfigurations D1, D2, . . . , D6 are shown in FIG. 9 as contours of0.25 intensity levels. For the 250 nm design, the source D6 is acombination of the diagonal quadrapole and a weak vertical dipole. Thisconfiguration is different from the one found in the Barouch paper,which looks like an annulus of 0.6 and 0.72 radii. Simulations along thehorizontal cutline a-a through the center of the cell show potentialsuperiority of D6 in delivering better image slopes. At the threshold0.3, we get the following intensity slopes (in 1/micron): 4.6, 3.3, 3.4,3.9, 4.0, 3.7, 3.8, and 3.4, while the best illumination in the Barouchpaper results in 4.3, 3.1, 3.3, 3.3, 3.4, 3.6, 3.6, and 3.4, which is onaverage 7% smaller. However, it is not completely clear whether thisdifference is mainly due to the difference in source representation, orcaused by the other factors, like optimization objective, defocussettings, different boundary conditions, etc.

With a decrease in feature size, the bright spots undergo non-trivialtopological and size transformations. From D6 to D5, vertical dipoleelements move to the periphery and then merge with the quad elements inD4. From D4 to D3 the quad elements stretch to the center and narrow; asecondary vertical quad emerges. D2 and D1 are 12-poles, with brightspots between σ=0.8 and σ=0.68. D1 poles are rotated 15° from D2 poles.

This example highlights the shortcomings of the contour-based sourceoptimization. While it is appropriate for shaping the predefined brightspots, it misses beneficial bright spots outside of the initial,predefined topology.

EXAMPLE 4 Contact Pattern for 0.11 Micron Design

The semi-dense contact pattern of this example is shown in FIG. 10.Contacts of size 110 nm are printed with a binary mask on a darkbackground, using λ=193 nm, NA=0.63, and σ=0.9. Optimizations areconducted with the balancing parameter γ=0.91. First, a selectiveweighting style where only densely placed contacts are heavilyaccentuated with a weight of 64 is used, while the weighting layer doesnot cover the isolated contact in the middle. The printing fidelity ofthis contact is left to be improved by the proximity correctionprocedure, which is an easy task, because plenty of room is left evenfor large corrections. The resulting source, E1, can be characterized asa horizontal dipole with “whiskers” of bright archels at x=±0.5. Withincreased smoothing at γ=0.56, the dipoles grow in size and join the“whiskers” (this source is not shown). Second, the optical correctionprocedure is incorporated into the optimization loop so that the sourceand mask are optimized simultaneously. Looping over the sourceoptimizations and the mask corrections results in a convergent process.The resulting solution delivers the best image fidelity by means of bothsource and mask modifications. When contact holes are not weighted, theiterations yield source configuration E2, which has larger and rounderdipole elements than E1. When the weighting and corrections are usedtogether the iteration procedure converges to the source E3, withdistinguished V-shaped dipoles.

EXAMPLE 5 Quadrapole Optimization

Pole smoothing of quad illuminations was proposed in the Smith,Zavyalova paper to mitigate proximity effects. It was done outside ofthe source optimization procedure using the Gaussian distribution ofintensity. The optimization problem of equation 46 naturallyincorporates the smoothing constraints and thus can be used to smooththe illumination poles in an optimal way. In this example, the processconditions and pattern from the SRAM design I example are reused, butchanged the optimization domain to a diagonal quadrapole between circlesof σ=0.47 and σ=0.88. The Sobolev norm parameters in equation 46 arechosen to be α₀=0, α₁=0, α₂=1, so that the second derivative of thesource intensity serves as a smoothing factor.

Results of optimization are shown in FIG. 11 for six values of thebalancing parameter γ: 0, 0.04, 0.14, 0.24, 0.38, and 0.56. Initialuniformly distributed intensity of the source map D1 becomes non-uniformwith introduction of the fidelity objective as a small portion of thewhole objective function in D2. The inner portions of the poles becomedarker and then portions of the poles completely disappear in D3.Sources D4, D5, and D6 are too dark to be recommended as smoothedreplacements of the original illumination. A reasonable compromisebetween smoothness and fidelity is source D2, which is still aquadrapole but the light inside the poles is redistributed in a way thatbenefits the SRAM printing fidelity.

As can be seen, the present invention provides a uniform, norm-basedapproach to the classification of optimization objectives. The imagefidelity in the frequency and in the space domains and expressed throughdifferent functional norms are compared. The Sobolev norm is proposedfor the throughput side-constraint. In one embodiment, the weightedEuclidean image fidelity is proposed as a main optimization objectiveand the averaging techniques to account for the defocus latitude arediscussed. In addition, in one embodiment the off-focus optimization isadopted to save run time. A strict formulation of the sourceoptimization problem is described and one solution method is developedas a reduction to a sequence of NNLS problems. Comparing the results forsimple periodic structures indicates the methods of the presentinvention are in good agreement with the previously found archel-basedresults. With the present invention, new advantageous source designs arefound, which demonstrate importance of the comprehensive optimizationapproach. Twelve- and ten-pole source shapes are found as the optimumsource configurations for the SRAM structures. In some situations, theadvantages of the pixel-based optimization over the contour-based aredemonstrated. The selective and the uniform weighting schemes for theimage fidelity are proposed. The iterative source/mask optimization isproposed, which alternates OPC and source optimization steps. Finally,the source can be smoothed by optimizing to print certain importantshapes.

While the preferred embodiment of the invention has been illustrated anddescribed, it will be appreciated that various changes can be madetherein without departing from the scope of the invention. It istherefore intended that the scope of the invention be determined fromthe following claims and equivalents thereof.

1. A method of optimizing a light source to create a desired pattern offeatures on a wafer with a photolithographic process, comprising:receiving at least a portion of a layout database and selecting adesired pattern of features to be created on a wafer; relating a numberof pixel intensities in a light source and the contribution of thepixels in the light source to a point on the wafer to the desiredpattern of features using a mathematical relationship; and determiningthe pixel intensities of the light source using the mathematicalrelationship such that upon simultaneous illumination of the pixels attheir determined pixel intensities, the errors between a pattern offeatures that will be created on the wafer and the desired pattern offeatures are minimized.
 2. The method of claim 1, wherein one or moreconstraints are placed on the mathematical relationship.
 3. The methodof claim 2, wherein the one or more constraints include limiting theoptical power of the combined pixel intensities.
 4. The method of claim2, wherein the one or more constraints include limiting the pixelintensities such that each pixel having a non-zero intensity is adjacentanother pixel having a non-zero intensity.
 5. The method of claim 2,wherein the one or more constraints include limiting the pixelintensities such that the intensities of adjacent pixels do not vary bymore than a predefined amount.
 6. The method of claim 1, wherein thedesired pattern of features is selected by determining a pattern offeatures that occurs at two or more locations on the wafer and theintensities of the pixels in the light source is determined byoptimizing the light source pixel intensities to produce a minimum errorin the selected pattern of features and the features that will becreated on a wafer.
 7. The method of claim 1, wherein the desiredpattern of features is selected by determining a pattern of featuresthat is repeated.
 8. The method of claim 1, wherein the desired patternof features is selected by determining a pattern of features that occursin an array.
 9. The method of claim 1, further comprising selectingareas of the desired pattern of features and weighting the selectedareas in the mathematical relationship such that the errors at theselected areas are minimized.
 10. The method of claim 1, in which themathematical relationship is a matrix equation.
 11. The method of claim1, further comprising creating an optical element that simultaneouslyproduces the pixels having the determined light source intensities whenilluminated by a light source of a photolithographic system.
 12. Themethod of claim 11, wherein the optical element is a diffractive opticalelement.
 13. A computer readable medium containing a number ofinstructions that are executable by a computer to perform the method ofclaim
 1. 14. A diffractive optical element that produces a distributionof illumination light for a photolithographic process that is producedin accordance with the method of claim
 1. 15. A method of preparing afile that defines a desired pattern of features to be created by aphotolithographic process, comprising: receiving all or a portion of alayout database from which a target pattern of features to be createdvia a photolithographic process is created; correcting the features ofthe layout database with a resolution enhancement technique such that anerror between a pattern of features that will be printed on a wafer andthe target pattern of features is minimized; using the correctedfeatures to optimize a distribution of light from an illumination lightsource such that when the corrected features are illuminated by thelight source, the error between the pattern of features that will becreated on a wafer and the target pattern of features is minimized. 16.The method of claim 15, further comprising: further correcting thefeatures with a resolution enhancement technique assuming illuminationwith an optimized distribution of light; and further optimizing thedistribution of light using the further corrected features so that theerror between the pattern of features that will be created on a waferand the target pattern of features is minimized.
 17. A computer readablemedium containing a number of instructions that are executable by acomputer to perform the method of claim
 15. 18. A diffractive opticalelement that produces a distribution of illumination light for aphotolithographic process that is produced in accordance with the methodof claim
 15. 19. A method of preparing a file that defines a desiredpattern of features to be created by a photolithographic process,comprising: receiving all or a portion of a layout database from which atarget pattern of features to be created via a photolithographic processis created; optimizing a distribution of light from an illuminationlight source such that when features are illuminated by the illuminationlight source, the error between a pattern of features that will becreated on a wafer and the target pattern of features is minimized; andcorrecting the features of the layout database with a resolutionenhancement technique assuming illumination with the optimizeddistribution of light from the illumination light source such that anerror between a pattern of features that will be printed on a wafer andthe target pattern of features is minimized.
 20. The method of claim 19,further comprising: further optimizing the distribution of light usingthe corrected features so that the error between the pattern of featuresthat will be created on a wafer and the target pattern of features isminimized; and further correcting the corrected features with aresolution enhancement technique assuming illumination with the furtheroptimized distribution of light.
 21. A computer readable mediumcontaining a number of instructions that are executable by a computer toperform the method of claim
 19. 22. A diffractive optical element thatproduces a distribution of illumination light for a photolithographicprocess that is produced in accordance with the method of claim 19.